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Theorem lautcnv 40073
Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
Hypothesis
Ref Expression
lautcnv.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautcnv ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)

Proof of Theorem lautcnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 lautcnv.i . . . 4 𝐼 = (LAut‘𝐾)
31, 2laut1o 40068 . . 3 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4 f1ocnv 6861 . . 3 (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
53, 4syl 17 . 2 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6 eqid 2735 . . . 4 (le‘𝐾) = (le‘𝐾)
71, 6, 2lautcnvle 40072 . . 3 (((𝐾𝑉𝐹𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
87ralrimivva 3200 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
91, 6, 2islaut 40066 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
109adantr 480 . 2 ((𝐾𝑉𝐹𝐼) → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
115, 8, 10mpbir2and 713 1 ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  ccnv 5688  1-1-ontowf1o 6562  cfv 6563  Basecbs 17245  lecple 17305  LAutclaut 39968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-laut 39972
This theorem is referenced by:  ldilcnv  40098
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